Make particles clump together and flow in a specific direction
This is campfire example.

To achieve this effect add two force fields.

First one: Shift+A > Force Field > Turbulence

Then change Noise to 5.



Second one: Shift+A > Force Field > Vortex

Change Noise to 2 and Strenght to 0.1.



Blend file:



Final toughts:

I don't know your full setup so some values may vary. You can try change position of force fields and strength/noise values.

Edit:

Based on provided blend file I've change all Force Fields to Force with negative strenght and noise. Also I've zeroed Velocity > Emitter Oject > X and Random.

This is how it looks with Motion Blur.



Updated blend file: